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Mirrors > Home > ILE Home > Th. List > cc2lem | Unicode version |
Description: Lemma for cc2 7082. (Contributed by Jim Kingdon, 27-Apr-2024.) |
Ref | Expression |
---|---|
cc2.cc | CCHOICE |
cc2.a | |
cc2.m | |
cc2lem.a | |
cc2lem.g |
Ref | Expression |
---|---|
cc2lem |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cc2.cc | . . 3 CCHOICE | |
2 | vex 2689 | . . . . . . . 8 | |
3 | 2 | snex 4109 | . . . . . . 7 |
4 | cc2.a | . . . . . . . 8 | |
5 | funfvex 5438 | . . . . . . . . 9 | |
6 | 5 | funfni 5223 | . . . . . . . 8 |
7 | 4, 6 | sylan 281 | . . . . . . 7 |
8 | xpexg 4653 | . . . . . . 7 | |
9 | 3, 7, 8 | sylancr 410 | . . . . . 6 |
10 | cc2lem.a | . . . . . 6 | |
11 | 9, 10 | fmptd 5574 | . . . . 5 |
12 | sneq 3538 | . . . . . . . . . 10 | |
13 | fveq2 5421 | . . . . . . . . . 10 | |
14 | 12, 13 | xpeq12d 4564 | . . . . . . . . 9 |
15 | simprr 521 | . . . . . . . . 9 | |
16 | vex 2689 | . . . . . . . . . . 11 | |
17 | 16 | snex 4109 | . . . . . . . . . 10 |
18 | 4 | adantr 274 | . . . . . . . . . . 11 |
19 | funfvex 5438 | . . . . . . . . . . . 12 | |
20 | 19 | funfni 5223 | . . . . . . . . . . 11 |
21 | 18, 15, 20 | syl2anc 408 | . . . . . . . . . 10 |
22 | xpexg 4653 | . . . . . . . . . 10 | |
23 | 17, 21, 22 | sylancr 410 | . . . . . . . . 9 |
24 | 10, 14, 15, 23 | fvmptd3 5514 | . . . . . . . 8 |
25 | 24 | eqeq2d 2151 | . . . . . . 7 |
26 | simpr 109 | . . . . . . . . . . 11 | |
27 | 10 | fvmpt2 5504 | . . . . . . . . . . 11 |
28 | 26, 9, 27 | syl2anc 408 | . . . . . . . . . 10 |
29 | 28 | adantrr 470 | . . . . . . . . 9 |
30 | 29 | eqeq1d 2148 | . . . . . . . 8 |
31 | 2 | snm 3643 | . . . . . . . . . 10 |
32 | fveq2 5421 | . . . . . . . . . . . . 13 | |
33 | 32 | eleq2d 2209 | . . . . . . . . . . . 12 |
34 | 33 | exbidv 1797 | . . . . . . . . . . 11 |
35 | cc2.m | . . . . . . . . . . . 12 | |
36 | 35 | adantr 274 | . . . . . . . . . . 11 |
37 | simprl 520 | . . . . . . . . . . 11 | |
38 | 34, 36, 37 | rspcdva 2794 | . . . . . . . . . 10 |
39 | xp11m 4977 | . . . . . . . . . 10 | |
40 | 31, 38, 39 | sylancr 410 | . . . . . . . . 9 |
41 | 2 | sneqr 3687 | . . . . . . . . . 10 |
42 | 41 | adantr 274 | . . . . . . . . 9 |
43 | 40, 42 | syl6bi 162 | . . . . . . . 8 |
44 | 30, 43 | sylbid 149 | . . . . . . 7 |
45 | 25, 44 | sylbid 149 | . . . . . 6 |
46 | 45 | ralrimivva 2514 | . . . . 5 |
47 | dff13 5669 | . . . . 5 | |
48 | 11, 46, 47 | sylanbrc 413 | . . . 4 |
49 | f1f1orn 5378 | . . . . 5 | |
50 | omex 4507 | . . . . . 6 | |
51 | 50 | f1oen 6653 | . . . . 5 |
52 | ensym 6675 | . . . . 5 | |
53 | 49, 51, 52 | 3syl 17 | . . . 4 |
54 | 48, 53 | syl 14 | . . 3 |
55 | 9 | ralrimiva 2505 | . . . . . . . . 9 |
56 | 10 | fnmpt 5249 | . . . . . . . . 9 |
57 | 55, 56 | syl 14 | . . . . . . . 8 |
58 | 57 | adantr 274 | . . . . . . 7 |
59 | fnfun 5220 | . . . . . . 7 | |
60 | 58, 59 | syl 14 | . . . . . 6 |
61 | simpr 109 | . . . . . 6 | |
62 | elrnrexdm 5559 | . . . . . 6 | |
63 | 60, 61, 62 | sylc 62 | . . . . 5 |
64 | simpll 518 | . . . . . . 7 | |
65 | simprl 520 | . . . . . . . 8 | |
66 | fndm 5222 | . . . . . . . . 9 | |
67 | 64, 57, 66 | 3syl 17 | . . . . . . . 8 |
68 | 65, 67 | eleqtrd 2218 | . . . . . . 7 |
69 | 35 | adantr 274 | . . . . . . . . . . 11 |
70 | 34, 69, 26 | rspcdva 2794 | . . . . . . . . . 10 |
71 | eleq1 2202 | . . . . . . . . . . 11 | |
72 | 71 | cbvexv 1890 | . . . . . . . . . 10 |
73 | 70, 72 | sylib 121 | . . . . . . . . 9 |
74 | vsnid 3557 | . . . . . . . . . . 11 | |
75 | simpr 109 | . . . . . . . . . . 11 | |
76 | opelxpi 4571 | . . . . . . . . . . 11 | |
77 | 74, 75, 76 | sylancr 410 | . . . . . . . . . 10 |
78 | eleq1 2202 | . . . . . . . . . . 11 | |
79 | 78 | spcegv 2774 | . . . . . . . . . 10 |
80 | 77, 77, 79 | sylc 62 | . . . . . . . . 9 |
81 | 73, 80 | exlimddv 1870 | . . . . . . . 8 |
82 | 28 | eleq2d 2209 | . . . . . . . . 9 |
83 | 82 | exbidv 1797 | . . . . . . . 8 |
84 | 81, 83 | mpbird 166 | . . . . . . 7 |
85 | 64, 68, 84 | syl2anc 408 | . . . . . 6 |
86 | simprr 521 | . . . . . . . 8 | |
87 | 86 | eleq2d 2209 | . . . . . . 7 |
88 | 87 | exbidv 1797 | . . . . . 6 |
89 | 85, 88 | mpbird 166 | . . . . 5 |
90 | 63, 89 | rexlimddv 2554 | . . . 4 |
91 | 90 | ralrimiva 2505 | . . 3 |
92 | 1, 54, 91 | ccfunen 7079 | . 2 |
93 | vex 2689 | . . . . . . . 8 | |
94 | funfvex 5438 | . . . . . . . . . 10 | |
95 | 94 | funfni 5223 | . . . . . . . . 9 |
96 | 57, 95 | sylan 281 | . . . . . . . 8 |
97 | fvexg 5440 | . . . . . . . 8 | |
98 | 93, 96, 97 | sylancr 410 | . . . . . . 7 |
99 | 2ndexg 6066 | . . . . . . 7 | |
100 | 98, 99 | syl 14 | . . . . . 6 |
101 | 100 | ralrimiva 2505 | . . . . 5 |
102 | cc2lem.g | . . . . . 6 | |
103 | 102 | fnmpt 5249 | . . . . 5 |
104 | 101, 103 | syl 14 | . . . 4 |
105 | 104 | adantr 274 | . . 3 |
106 | simpr 109 | . . . . . 6 | |
107 | fveq2 5421 | . . . . . . . . . 10 | |
108 | id 19 | . . . . . . . . . 10 | |
109 | 107, 108 | eleq12d 2210 | . . . . . . . . 9 |
110 | simplrr 525 | . . . . . . . . 9 | |
111 | fnfvelrn 5552 | . . . . . . . . . . 11 | |
112 | 57, 111 | sylan 281 | . . . . . . . . . 10 |
113 | 112 | adantlr 468 | . . . . . . . . 9 |
114 | 109, 110, 113 | rspcdva 2794 | . . . . . . . 8 |
115 | 28 | eleq2d 2209 | . . . . . . . . 9 |
116 | 115 | adantlr 468 | . . . . . . . 8 |
117 | 114, 116 | mpbid 146 | . . . . . . 7 |
118 | xp2nd 6064 | . . . . . . 7 | |
119 | 117, 118 | syl 14 | . . . . . 6 |
120 | 102 | fvmpt2 5504 | . . . . . 6 |
121 | 106, 119, 120 | syl2anc 408 | . . . . 5 |
122 | 121, 119 | eqeltrd 2216 | . . . 4 |
123 | 122 | ralrimiva 2505 | . . 3 |
124 | 50 | a1i 9 | . . . . . 6 |
125 | fnex 5642 | . . . . . 6 | |
126 | 104, 124, 125 | syl2anc 408 | . . . . 5 |
127 | fneq1 5211 | . . . . . . 7 | |
128 | fveq1 5420 | . . . . . . . . 9 | |
129 | 128 | eleq1d 2208 | . . . . . . . 8 |
130 | 129 | ralbidv 2437 | . . . . . . 7 |
131 | 127, 130 | anbi12d 464 | . . . . . 6 |
132 | 131 | spcegv 2774 | . . . . 5 |
133 | 126, 132 | syl 14 | . . . 4 |
134 | 133 | adantr 274 | . . 3 |
135 | 105, 123, 134 | mp2and 429 | . 2 |
136 | 92, 135 | exlimddv 1870 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 wex 1468 wcel 1480 wral 2416 wrex 2417 cvv 2686 csn 3527 cop 3530 class class class wbr 3929 cmpt 3989 com 4504 cxp 4537 cdm 4539 crn 4540 wfun 5117 wfn 5118 wf 5119 wf1 5120 wf1o 5122 cfv 5123 c2nd 6037 cen 6632 CCHOICEwacc 7077 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-iinf 4502 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-2nd 6039 df-er 6429 df-en 6635 df-cc 7078 |
This theorem is referenced by: cc2 7082 |
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